Let A be a set; (Xas Ta)ae be an indexed family of topological spaces; and let...

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Let A be a set; (Xas Ta)ae be an indexed family of topological spaces; and let {fa}ae be the indexed family of functions fa: A Xa. (a) Show that there is a unique coarsest topology T on A with respect to which each fa is continuous. (b) Show that the collection S = {f(UB) | B J and Us T} forms a subbasis for T. (c) Let f: AIX be defined by aEJ f(a) = (fa(a))aEJ. Show that if U ET, then f(U) is an open subset of f(A). Let A be a set; (Xas Ta)ae be an indexed family of topological spaces; and let {fa}ae be the indexed family of functions fa: A Xa. (a) Show that there is a unique coarsest topology T on A with respect to which each fa is continuous. (b) Show that the collection S = {f(UB) | B J and Us T} forms a subbasis for T. (c) Let f: AIX be defined by aEJ f(a) = (fa(a))aEJ. Show that if U ET, then f(U) is an open subset of f(A).

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